In this paper, we study the scaling limit of the pinning model in correlated Gaussian environment. The tail probability of the underlying renewal process of the model has a polynomial decay with exponent $\alpha>0$. The covariance of the Gaussian environment $\{\omega_n\}_{n\in\mathbb N}$ is given by $\text{Cov}_{\mathbb P}(\omega_n,\omega_m)\sim |n-m|^{2H-2}$ with $H\in(0,1)$. Assuming $\alpha\in(0,\frac12]$, $H\in(\frac12,1)$ and $\alpha+2H>2$, we show that the partition function of the disordered pinning model, under the appropriate scaling, converges in distribution to the $L^1$-solution of the fractional stochastic heat equation driven by Gaussian noise correlated in time and localized at the origin. In particular, it is known that the solution is not $L^2$-integrable when $\alpha<\frac12$.
We establish free-energy fluctuation limits for the Ising Sherrington--Kirkpatrick model in the nonzero parts of its critical window. For fixed $b\ne0$ and $\beta_N=1+bN^{-1/3}\sqrt{\log N}$, our main Gaussian orthogonal ensemble (GOE) result is \[ \sqrt{\frac6{\log N}}\left(F_{N,\beta_N}-N\,\mathrm{FE}(\beta_N)+\frac{...
We study the Ising model on $\mathbb{Z}^d$ with $d\geq 4$ and derive near-critical bounds on the truncated two-point function $\langle\sigma_0;\sigma_x\rangle_{\beta,h} := \langle\sigma_0\sigma_x\rangle_{\beta,h} - \langle\sigma_0\rangle_{\beta,h}\langle\sigma_x\rangle_{\beta,h}$ at parameters $\beta\leq\beta_c$ and $h...
We present a comprehensive Monte Carlo study of two-dimensional bond percolation with algebraically decaying connection probabilities $p(r)\propto 1/r^{2+\sigma}$, establishing the universality diagram in the long-range (LR) regime for $\sigma\le2$. Using the event-based ensemble method, we simulate systems with linear...
Zi-Yu Liu, Tian-Ning Xiao, Zhi-Jie Fan et al.· 2 citations
This paper establishes a mesoscopic central limit theorem for linear statistics of $\beta$-ensembles or log-gas, as the dimension $N\to\infty$, in the temperature regime $1/N\ll\beta(N)\le 1$. For simplicity, we assume that the potential is one-cut regular and analytic. In this regime, the size of the fluctuations depe...
Charlie Dworaczek Guera, Gaultier Lambert, Luke Peilen· 0 citations
We prove cutoff with an $O(1)$ window for the continuous-time heat-bath Glauber dynamics of the ferromagnetic $q$-state Potts model on the discrete torus $\Lambda_n=(\mathbb Z/n\mathbb Z)^d$ at sufficiently high temperature. For every fixed $d\ge2$ and $q\ge3$, there exists $\beta_0=\beta_0(d,q)>0$ such that, for $0<\b...
We study the smallest nonzero eigenvalue of the sample correlation matrix $\mathbf{R}_n$ formed from a $p_n \times n$ data matrix with i.i.d. real entries $\xi$ of mean zero and unit variance, in the high-dimensional regime $p_n / n \to \phi \in (0, \infty) \setminus \{1\}$. In the tall regime $\phi>1$, we prove almost...
Ze-Qin Lin, Guamgming Pan, Hao-Zhu Zhao et al.· 1 citation
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